228 ANALYTICAL MECHANICS 5. In the second illustrative problem suppose there is a resistance to the motion of the falling body proportional to its velocity. 6. A flywheel making 400 revolutions per minute is brought to rest in 3 minutes by means of friction brakes applied to it. Find the angular inertia of the wheel and axle if the total brake-shoe force applied is 500 pounds and the diameter of the flywheel is 10 feet. 7. In the preceding problem find the total number of revolutions made after the brakes were applied. 8. A flywheel is brought to rest by means of brakes applied at the axle. If the combined angular inertia of the flywheel and the axle is 50,000 gm. cm.2 and the diameter of the axle 20 cm., find the force which must be applied on the brakes in order to bring the flywheel to rest within 5 minutes, the initial angular velocity being 30 radians per sec. 9. A flywheel is stopped by fluid friction. The resisting torque due to the friction is proportional to the angular velocity. Discuss the motion. 10. The flywheel of a gyroscope is rotated by applying a force to a string wound around the axle. Discuss the motion, supposing the tension of the string to be proportional to the length of the string unwound. MOTION OF A RIGID BODY ABOUT INSTANTANEOUS AXES. 188. Uniplanar Motion. — It was shown on p. 31 that uniplanar motion may be considered as a motion of pure rotation at each instant of the motion. Since the torque and energy equations hold good at each instant of the motion they can be applied to a rigid body in uniplanar motion as if the in- o*x stantaneous axis were fixed at the instant considered. Therefore uniplanar motion may be discussed in the same way as motion about a fixed axis. 189. Instantaneous Axis. — If at any instant the velocities of two points of a rigid body are FlQ 114 known the position of the instantaneous axis may be found in the following manner: Let P and Q, Fig. 114, be two points which lie in a plane parallel to the guide plane, and the velocities of which are